A332350 -id:A332350 - cp's OEIS frontend

A332369 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of quadrilateral cells in the partition for m >= n >= 2.

3, 6, 9, 11, 18, 35, 18, 27, 52, 77, 27, 42, 81, 122, 191, 38, 57, 108, 159, 248, 321, 51, 78, 147, 216, 335, 436, 591, 66, 99, 186, 273, 424, 551, 746, 941, 83, 126, 235, 346, 537, 698, 943, 1190, 1503, 102, 153, 284, 415, 642, 829, 1118, 1407, 1776, 2097, 123, 186, 345, 504, 777, 1002, 1349, 1696, 2139, 2528, 3047

Offset: 2

N. J. A. Sloane, Feb 12 2020

			Triangle begins:
3,
6, 9,
11, 18, 35,
18, 27, 52, 77,
27, 42, 81, 122, 191,
38, 57, 108, 159, 248, 321,
51, 78, 147, 216, 335, 436, 591,
66, 99, 186, 273, 424, 551, 746, 941,
83, 126, 235, 346, 537, 698, 943, 1190, 1503,...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3

Cf. A332350, A332352, A331781, A332367, A332371, A332372, A332374.

For main diagonal see A332370.

Maple
```
See A332367.
```

A332353 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 2, 4, 14, 24, 3, 6, 22, 38, 60, 4, 8, 30, 52, 82, 112, 5, 10, 40, 70, 112, 154, 212, 6, 12, 50, 88, 142, 196, 270, 344, 7, 14, 62, 110, 178, 246, 340, 434, 548, 8, 16, 74, 132, 214, 296, 410, 524, 662, 800, 9, 18, 88, 158, 258, 358, 498, 638, 808, 978, 1196

Offset: 1

N. J. A. Sloane, Feb 10 2020

This is the triangle in A332352, halved.

			Triangle begins:
0,
0, 0,
1, 2, 8,
2, 4, 14, 24,
3, 6, 22, 38, 60,
4, 8, 30, 52, 82, 112,
5, 10, 40, 70, 112, 154, 212,
6, 12, 50, 88, 142, 196, 270, 344,
7, 14, 62, 110, 178, 246, 340, 434, 548,
8, 16, 74, 132, 214, 296, 410, 524, 662, 800,
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. This sequence is f_2(m,n)/2.

The main diagonal is A177719.

Cf. A332350, A332352.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
for m from 1 to 12 do lprint(seq(VR(m,n,2)/2,n=1..m),); od:

A332353[m_,n_]:=Sum[If[GCD[i,j]==2,2(m-i)(n-j),0],{i,2,m-1,2},{j,2,n-1,2}]+If[n>2,m*n-2m,0]+If[m>2,m*n-2n,0];Table[A332353[m, n],{m,15},{n, m}] (* Paolo Xausa, Oct 18 2023 *)

A332361 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of vertices in the partition, for m >= n >= 1.

Original entry on oeis.org

3, 4, 6, 5, 9, 14, 6, 13, 22, 36, 7, 18, 31, 52, 76, 8, 24, 43, 74, 110, 160, 9, 31, 56, 97, 144, 210, 276, 10, 39, 72, 126, 188, 275, 363, 478, 11, 48, 89, 157, 235, 345, 456, 601, 756, 12, 58, 109, 193, 290, 427, 565, 745, 938, 1164, 13, 69, 130, 231, 347, 511, 675, 890, 1120, 1390, 1660

Offset: 1

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
3,
4, 6,
5, 9, 14,
6, 13, 22, 36,
7, 18, 31, 52, 76,
8, 24, 43, 74, 110, 160,
9, 31, 56, 97, 144, 210, 276,
10, 39, 72, 126, 188, 275, 363, 478,
11, 48, 89, 157, 235, 345, 456, 601, 756,
12, 58, 109, 193, 290, 427, 565, 745, 938, 1164,
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 13.
N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

Cf. A332350-A331360.

Main diagonal is A332362.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
ct3 := proc(m,n) local i; global VR;
if m=1 or n=1 then max(m,n) else VR(m,n,2)/2+m+n+1; fi; end; # A332354
ct4 := proc(m,n) local i; global VR;
if m=1 or n=1 then 0 else VR(m,n,1)/4-VR(m,n,2)/2-m/2-n/2-1; fi; end; # A332356
ct := (m,n) -> ct3(m,n) + ct4(m,n); # A332357
cte := proc(m,n) local i; global VR;
if m=1 or n=1 then 2*max(m,n)+1 else VR(m,n,1)/2-VR(m,n,2)/4+m+n; fi; end; # A332359
ctv := (m,n) -> cte(m,n) - ct(m,n) + 1; # A332361
for m from 1 to 12 do lprint([seq(ctv(m,n),n=1..m)]); od:

T(m,n) = A332359(m,n) - A332357(m,n) + 1 (Euler's formula).

A332363 Triangle read by rows: T(m,n) = number of unstable threshold functions (the function u_{0,1}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

Original entry on oeis.org

1, 2, 7, 3, 11, 19, 4, 18, 31, 51, 5, 24, 42, 69, 95, 6, 33, 59, 98, 135, 191, 7, 41, 74, 124, 172, 243, 311, 8, 52, 94, 158, 219, 310, 397, 507, 9, 62, 114, 191, 265, 376, 482, 615, 747, 10, 75, 138, 233, 325, 462, 593, 758, 921, 1135

Offset: 2

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
1,
2, 7,
3, 11, 19,
4, 18, 31, 51,
5, 24, 42, 69, 95,
6, 33, 59, 98, 135, 191,
7, 41, 74, 124, 172, 243, 311,
8, 52, 94, 158, 219, 310, 397, 507,
9, 62, 114, 191, 265, 376, 482, 615, 747,
10, 75, 138, 233, 325, 462, 593, 758, 921, 1135,
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 11.

Cf. A332350, A332352, A331781.

Main diagonal is A332364.

VQ := proc(m,n,q) local eps,a,i,j; eps := 10^(-6); a:=0;
for i from ceil(-m+eps) to floor(m-eps) do
for j from ceil(-n+eps) to floor(n-eps) do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
VS := proc(m,n) local a,i,j; a:=0;
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end; # A331781
u01:=(m,n) -> 2*VQ(m/2,n/2,1)+2-VS(m,n); # This sequence
for m from 2 to 12 do lprint([seq(u01(m,n),n=2..m)]); od:

A332365 Triangle read by rows: T(m,n) = number of threshold functions (the function u_{0,2}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

Original entry on oeis.org

3, 6, 13, 9, 21, 33, 12, 30, 49, 73, 15, 40, 66, 99, 133, 18, 51, 85, 130, 177, 237, 21, 63, 106, 164, 224, 301, 381, 24, 76, 130, 202, 277, 374, 475, 593, 27, 90, 154, 241, 331, 448, 570, 713, 857, 30, 105, 182, 287, 395, 538, 687, 862, 1039, 1261, 33, 121, 211, 335, 462, 632, 808, 1016, 1226, 1489, 1757

Offset: 2

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
3,
6, 13,
9, 21, 33,
12, 30, 49, 73,
15, 40, 66, 99, 133,
18, 51, 85, 130, 177, 237,
21, 63, 106, 164, 224, 301, 381,
24, 76, 130, 202, 277, 374, 475, 593,
27, 90, 154, 241, 331, 448, 570, 713, 857,
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 11.

Cf. A332350, A332352, A332363, A331781.

Main diagonal is A332366.

VQ := proc(m,n,q) local eps,a,i,j; eps := 10^(-6); a:=0;
for i from ceil(-m+eps) to floor(m-eps) do
for j from ceil(-n+eps) to floor(n-eps) do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
VS := proc(m,n) local a,i,j; a:=0;
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end; # A331781
u02:=(m,n) -> VQ(m,n,2)+2-2*VQ(m/2,n/2,1)+VS(m,n); # This sequence
for m from 2 to 12 do lprint([seq(u02(m,n),n=2..m)]); od:

cp's OEIS Frontend

A332369 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of quadrilateral cells in the partition for m >= n >= 2.

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A332353 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

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A332361 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of vertices in the partition, for m >= n >= 1.

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A332363 Triangle read by rows: T(m,n) = number of unstable threshold functions (the function u_{0,1}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

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Maple

A332365 Triangle read by rows: T(m,n) = number of threshold functions (the function u_{0,2}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

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Author

Keywords

Examples

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Programs

Maple

A332369 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of quadrilateral cells in the partition for m >= n >= 2.

A332361 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of vertices in the partition, for m >= n >= 1.